Firstly, denote by $P$ as the set of holomorphic functions that extend $f$, where $f: D \rightarrow \mathbb{C}$ is holomorphic, and $D$ is a domain in $\mathbb{C}$. We have to show that $P$ has a maximal element. This should involve invoking Zorn's Lemma, but I have no clue how to proceed.
Secondly, if we pick a maximal element from the previous part, and call it $g: D^{'}\rightarrow \mathbb{C}$. Now consider a point $a \in D^{'}$ and take the power series expansion centered around $a$, $f(z) = \sum_{i=0}^\infty c_n(z - a)^n.$ If we denote by $R_1$ the radius of convergence(Hadamard) for the power series, and define $R_2 = \text{sup}\{R \geq 0 \mid \mathbb{D}(a, R) \subseteq D^{'} \}$ (geometric radius). Then show that $R_1 \geq R_2$.
I'd be obliged if someone could sketch out the proofs for this.